Let $k$ be fixed and consider the sum $$F(k,n)=\sum_{p_1<p_2<\cdots<p_k\leq n~:~p_1 p_2\cdots p_k\leq n} \frac{1}{p_1 p_2 \cdots p_k}.$$ Are tight upper and lower bounds on $F(k,n)$ known as $n\rightarrow \infty$? One approximation to the sum might be $$\left( \sum_{p\leq n^{1/k}} \frac{1}{p}\right)^k\approx \left[\log \left (\frac{1}{k} \log n\right)\right]^k=\left[\log \log n -\log k\right]^k $$but I am unsure how good this approximation is.

Now, let $k$ vary very slowly with $n$, say $k=c \log \log n.$ Do the answer(s) (if known) change?